Acta Math. Acad. Paed. Nyíregyháziensis
15 (1999), 35-40
www.bgytf.hu/~amapn

On the limit of a sequence

Z. László and Z. Vörös

Abstract:

The object of this article is to examine the sequence

\begin{displaymath}
a_n={\displaystyle{\sum_{i=0}^{n} {n^i \over {i!}}} \over e^n}\end{displaymath}

well known from probability theory. We prove that the sequence is bounded, strictly monotonously decreasing, and $
\mathop {\hbox{lim}}_{n \to \infty}a_n={1 \over 2} \ .
$The last two statements are proved by analytical means. Finally, a modification and a generalization of (an) will be mentioned, and the sketch of a second analytical proof for the original limit will be given.

 

Mathematics Subject Classification. 26A12, 26A42, 40A05.

Key words and phrases. Limit, probability, analytical means.


Back to the Contents of vol 15